5 problems

Let $\Lambda$ be a Leech lattice in phase space of dimension $24$, with arbitrary fixed density greater than $1$. The Leech-lattice conjecture. There exists a Leech lattice that so…

Let $c\in\mathbb R^{+}$. For sufficiently large dimension and sufficiently small $\epsilon$, define … Here $N_{c,\epsilon(n)}$ is the prescribed number of geodesic balls and…

Let $\epsilon=\epsilon(n)$ be the radius of the geodesic balls, let $N_{c,\epsilon(n)}$ be the number of independently and uniformly selected centers used for the covering, and let…

Let $N(\epsilon,n)$ be the number of geodesic balls considered in the sphere-covering problem, and let $c_1>0$ denote an absolute constant independent of $\epsilon$ and $n$. The ex…

Let $\mathbf{H}^5$ be the 5-dimensional hyperbolic space, and consider the Coxeter tiling $[5,3,3,3,3]$ and the hyperball covering described in the paper. Least dense hyperball cov…